Twelfth International Workshop on Algebraic and Combinatorial Coding Theory September 5-11, 2010,Akademgorodok, Novosibirsk, Russia pp. 93–97 New arcs in PG(2, 17) and PG(2, 19) 1 Rumen Daskalov daskalov@tugab.bg Elena Metodieva metodieva@tugab.bg Department of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, BULGARIA Abstract. An (n, r)-arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. In this paper new (95, 7)-arc and (205, 13)-arc in PG(2,17) are constructed, as well as (243, 14)-arc and (264, 15)-arc in PG(2,19). 1 Introduction Let GF(q) denote the Galois field of q elements and V(3, q) be the vector space of row vectors of length three with entries in GF(q). Let PG(2, q) be the corresponding projective plane. The points (x 1 ,x 2 ,x 3 ) of PG(2,q) are the 1-dimensional subspaces of V(3,q). Subspaces of dimension two are called lines. The number of points and the number of lines in PG(2, q) is q 2 + q + 1. There are q + 1 points on every line and q + 1 lines through every point. Definition 1.1 An (n, r)-arc is a set of n points of a projective plane such that some r, but no r +1 of them, are collinear. The maximum size of an (n, r)-arc in PG(2, q) is denoted by m r (2,q). Definition 1.2 Let M be a set of points in any plane. An i-secant is a line meeting M in exactly i points. Define τ i as the number of i-secants to a set M . In terms of τ i the definition of an (n, r)-arc becomes Definition 1.3 An (n, r)-arc is a set of n points of a projective plane for which τ i ≥ 0 for i<r, τ r > 0 and τ i =0 when i>r. Definition 1.4 An (l, t)-blocking set S in PG(2, q) is a set of l points such that every line of PG(2, q) intersects S in at least t points, and there is a line intersecting S in exactly t points. Note that an (n, r)-arc is the complement of a (q 2 + q +1 − n, q +1 − r)-blocking set in a projective plane and conversely. The following two theorems are proved in [1] and [2] respectively. 1 This work was partially supported by the Ministry of Education and Science under con- tract in TU-Gabrovo.